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Full text of "Treatise On Analysis Vol-Ii"

326       XV   NORMED ALGEBRAS AND SPECTRAL THEORY

(use Holder's inequality). Deduce that the function /?i >Dp(/x) is continuous on the
right on ]0, + oo [. (Use the fact that this function is upper semicontinuous.) If
/x(X) = 1, then 0 < Dp(ft) <; 1 for 0 < p < + oo.

If p> 1  and if fe &p(fj,) and #e JS?e(/x) (where l/p+ l/q= 1) are positive
functions, then

 ft) ^ (D,(/p  ft))1"  (D,(0  ft))1".
If ft(X) = 1, then for each positive function /e -^(/x), we have

(b)    If Dp(/x) = 1 for some p> 0, show that ft is a representative measure for ^,
(Observe that, if veB is such that xW =  then x(exp(to/p)) = 1  for a^ f>0;
show that \&tv dp ^ 0 by making / tend to 0, and deduce that I &v dp, = 0.)

(c)    Conversely, if /x is a representative measure for #, then Dp(/x) =1 for all p ^ 1.
Show that Dp(ft) = 1 for all p > 0 if and only if ft is a Jensen measure for % (use
Section 13.11, Problem 12(e)).

(d)    Let/? be such that 1 ^p < 4- oo and Dp(/x) > 0. Show that there exists a function
h e &*(& (where (1/p) 4- (I/?) = 1) such that N9(/i) = 1 and

for all w e B. (Use the Hahn-Banach theorem, Proposition (13.17.1) and Problem 1
of (13.17).) If p > 1, the measure \h\q - p is a representative measure for ^.

11.   With the hypotheses and notation of Problem 8, for every function/^ 0 belonging
to ^(ft), put !(/) = exp[ I   log |/| dfj\ (with exp(- oo) = 0). A representative

measure /x for x is said to be hyperextremal if every representative measure for x
with base //. is necessarily equal to /x ; it is then an extremal point of the convex set
$(x) (Problem 9).

(a)    Let;? ^ 1. For a positive measure p of mass 1 on X to be a Jensen measure for
X, it is necessary and sufficient that Dp(/- p,) J> J^(/) for all functions/ ^ 0 belonging
to ^((j) .(To show that the condition is sufficient, consider the function (\u\ + e)~p,
where u e B and e > 0.)

(b)    Let p ^ 1 and let /x be a positive measure of mass 1 on X. Then the following
conditions are equivalent :

(a)   /x is a hyperextremal representative measure for x-

(ft)   ^ is a representative measure for ^, and Dp(/- ft) g J^(/) for all functions

/;> 0 belonging to ^^/x).
(y)   /x is a representative measure for x> anc^ fr each/e ^'(ft) such that/- ft

is a quasi-representative measure for # we have  l/Vft

 Wl/l).

Moreover, if these conditions are satisfied, then ft is a Jensen measure for #, and
D//- ft) = JM(/)for all^ ^ 1 and all functions/^ 0 belonging to JS?1^)-

(To show that (a) implies ($), prove that if ft is hyperextremal and p ^ 1 and
/;>0 belongs to -^"(ft), then (Dp(/p  ft))1/p g Dj(/- ft): for this, use Problem 10(d).
Then use Problem 10(a) to deduce that iff,g are ;>0 and belong to -^(ft), then
&i(fff ' ft) ^ Di(/- ft)D!^  /x). Conclude that, for all functions /g> 0 in ^(ft), we).n particular, the image of C under the